Dimension of torus one-point modular-covariance solution spaces

Determine whether the space of modular-covariance solutions for torus one-point correlators with nontrivial abelian monodromy has dimension floor (r_1^2+9)/6+delta_{r_1 equiv 0 mod 6}-2 for primary fields with r_1 in the positive integers and channel phase theta satisfying theta_1=theta^{12}.

Background

The paper extends critical loop models to torus one-point correlators with abelian monodromies. Modular covariance imposes the phase relation theta_1=theta^{12}, and the conjecture gives a proposed finite dimension for the corresponding space of solutions.

In the trivial-phase limit, the proposed dimension differs from a previously tested counting formula by two units after diagonal fields are excluded. The paper specifically notes that the reason for this additional discrepancy is unresolved, while numerical checks are presented as evidence for the conjecture.